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Annotation of src/lib/libm/src/e_pow.c, Revision 1.15

1.1       jtc         1: /* @(#)e_pow.c 5.1 93/09/24 */
                      2: /*
                      3:  * ====================================================
                      4:  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
                      5:  *
                      6:  * Developed at SunPro, a Sun Microsystems, Inc. business.
                      7:  * Permission to use, copy, modify, and distribute this
1.11      simonb      8:  * software is freely granted, provided that this notice
1.1       jtc         9:  * is preserved.
                     10:  * ====================================================
                     11:  */
1.3       jtc        12:
1.10      lukem      13: #include <sys/cdefs.h>
1.7       jtc        14: #if defined(LIBM_SCCS) && !defined(lint)
1.15    ! lukem      15: __RCSID("$NetBSD: e_pow.c,v 1.14 2008/04/25 22:21:53 christos Exp $");
1.3       jtc        16: #endif
1.1       jtc        17:
                     18: /* __ieee754_pow(x,y) return x**y
                     19:  *
                     20:  *                   n
                     21:  * Method:  Let x =  2   * (1+f)
                     22:  *     1. Compute and return log2(x) in two pieces:
                     23:  *             log2(x) = w1 + w2,
                     24:  *        where w1 has 53-24 = 29 bit trailing zeros.
1.13      drochner   25:  *     2. Perform y*log2(x) = n+y' by simulating multi-precision
1.1       jtc        26:  *        arithmetic, where |y'|<=0.5.
                     27:  *     3. Return x**y = 2**n*exp(y'*log2)
                     28:  *
                     29:  * Special cases:
                     30:  *     1.  (anything) ** 0  is 1
                     31:  *     2.  (anything) ** 1  is itself
                     32:  *     3.  (anything) ** NAN is NAN
                     33:  *     4.  NAN ** (anything except 0) is NAN
                     34:  *     5.  +-(|x| > 1) **  +INF is +INF
                     35:  *     6.  +-(|x| > 1) **  -INF is +0
                     36:  *     7.  +-(|x| < 1) **  +INF is +0
                     37:  *     8.  +-(|x| < 1) **  -INF is +INF
                     38:  *     9.  +-1         ** +-INF is NAN
                     39:  *     10. +0 ** (+anything except 0, NAN)               is +0
                     40:  *     11. -0 ** (+anything except 0, NAN, odd integer)  is +0
                     41:  *     12. +0 ** (-anything except 0, NAN)               is +INF
                     42:  *     13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
                     43:  *     14. -0 ** (odd integer) = -( +0 ** (odd integer) )
                     44:  *     15. +INF ** (+anything except 0,NAN) is +INF
                     45:  *     16. +INF ** (-anything except 0,NAN) is +0
                     46:  *     17. -INF ** (anything)  = -0 ** (-anything)
                     47:  *     18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
                     48:  *     19. (-anything except 0 and inf) ** (non-integer) is NAN
                     49:  *
                     50:  * Accuracy:
                     51:  *     pow(x,y) returns x**y nearly rounded. In particular
                     52:  *                     pow(integer,integer)
1.11      simonb     53:  *     always returns the correct integer provided it is
1.1       jtc        54:  *     representable.
                     55:  *
                     56:  * Constants :
1.11      simonb     57:  * The hexadecimal values are the intended ones for the following
                     58:  * constants. The decimal values may be used, provided that the
                     59:  * compiler will convert from decimal to binary accurately enough
1.1       jtc        60:  * to produce the hexadecimal values shown.
                     61:  */
                     62:
1.4       jtc        63: #include "math.h"
                     64: #include "math_private.h"
1.1       jtc        65:
1.11      simonb     66: static const double
1.1       jtc        67: bp[] = {1.0, 1.5,},
                     68: dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
                     69: dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
                     70: zero    =  0.0,
                     71: one    =  1.0,
                     72: two    =  2.0,
                     73: two53  =  9007199254740992.0,  /* 0x43400000, 0x00000000 */
                     74: huge   =  1.0e300,
                     75: tiny    =  1.0e-300,
                     76:        /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
                     77: L1  =  5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
                     78: L2  =  4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
                     79: L3  =  3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
                     80: L4  =  2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
                     81: L5  =  2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
                     82: L6  =  2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
                     83: P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
                     84: P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
                     85: P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
                     86: P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
                     87: P5   =  4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
                     88: lg2  =  6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
                     89: lg2_h  =  6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
                     90: lg2_l  = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
                     91: ovt =  8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
                     92: cp    =  9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
                     93: cp_h  =  9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
                     94: cp_l  = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
                     95: ivln2    =  1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
                     96: ivln2_h  =  1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
                     97: ivln2_l  =  1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
                     98:
1.12      wiz        99: double
                    100: __ieee754_pow(double x, double y)
1.1       jtc       101: {
                    102:        double z,ax,z_h,z_l,p_h,p_l;
1.14      christos  103:        double yy1,t1,t2,r,s,t,u,v,w;
1.5       jtc       104:        int32_t i,j,k,yisint,n;
                    105:        int32_t hx,hy,ix,iy;
                    106:        u_int32_t lx,ly;
1.1       jtc       107:
1.4       jtc       108:        EXTRACT_WORDS(hx,lx,x);
                    109:        EXTRACT_WORDS(hy,ly,y);
1.1       jtc       110:        ix = hx&0x7fffffff;  iy = hy&0x7fffffff;
                    111:
                    112:     /* y==zero: x**0 = 1 */
1.11      simonb    113:        if((iy|ly)==0) return one;
1.1       jtc       114:
                    115:     /* +-NaN return x+y */
                    116:        if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
1.11      simonb    117:           iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
                    118:                return x+y;
1.1       jtc       119:
                    120:     /* determine if y is an odd int when x < 0
                    121:      * yisint = 0      ... y is not an integer
                    122:      * yisint = 1      ... y is an odd int
                    123:      * yisint = 2      ... y is an even int
                    124:      */
                    125:        yisint  = 0;
1.11      simonb    126:        if(hx<0) {
1.1       jtc       127:            if(iy>=0x43400000) yisint = 2; /* even integer y */
                    128:            else if(iy>=0x3ff00000) {
                    129:                k = (iy>>20)-0x3ff;        /* exponent */
                    130:                if(k>20) {
                    131:                    j = ly>>(52-k);
1.15    ! lukem     132:                    if((uint32_t)(j<<(52-k))==ly) yisint = 2-(j&1);
1.1       jtc       133:                } else if(ly==0) {
                    134:                    j = iy>>(20-k);
                    135:                    if((j<<(20-k))==iy) yisint = 2-(j&1);
                    136:                }
1.11      simonb    137:            }
                    138:        }
1.1       jtc       139:
                    140:     /* special value of y */
1.11      simonb    141:        if(ly==0) {
1.1       jtc       142:            if (iy==0x7ff00000) {       /* y is +-inf */
                    143:                if(((ix-0x3ff00000)|lx)==0)
                    144:                    return  y - y;      /* inf**+-1 is NaN */
                    145:                else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
                    146:                    return (hy>=0)? y: zero;
                    147:                else                    /* (|x|<1)**-,+inf = inf,0 */
                    148:                    return (hy<0)?-y: zero;
1.11      simonb    149:            }
1.1       jtc       150:            if(iy==0x3ff00000) {        /* y is  +-1 */
                    151:                if(hy<0) return one/x; else return x;
                    152:            }
                    153:            if(hy==0x40000000) return x*x; /* y is  2 */
                    154:            if(hy==0x3fe00000) {        /* y is  0.5 */
                    155:                if(hx>=0)       /* x >= +0 */
1.11      simonb    156:                return __ieee754_sqrt(x);
1.1       jtc       157:            }
                    158:        }
                    159:
                    160:        ax   = fabs(x);
                    161:     /* special value of x */
                    162:        if(lx==0) {
                    163:            if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
                    164:                z = ax;                 /*x is +-0,+-inf,+-1*/
                    165:                if(hy<0) z = one/z;     /* z = (1/|x|) */
                    166:                if(hx<0) {
                    167:                    if(((ix-0x3ff00000)|yisint)==0) {
                    168:                        z = (z-z)/(z-z); /* (-1)**non-int is NaN */
1.11      simonb    169:                    } else if(yisint==1)
1.1       jtc       170:                        z = -z;         /* (x<0)**odd = -(|x|**odd) */
                    171:                }
                    172:                return z;
                    173:            }
                    174:        }
1.11      simonb    175:
1.13      drochner  176:        n = (hx>>31)+1;
                    177:
1.1       jtc       178:     /* (x<0)**(non-int) is NaN */
1.13      drochner  179:        if((n|yisint)==0) return (x-x)/(x-x);
                    180:
                    181:        s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
                    182:        if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
1.1       jtc       183:
                    184:     /* |y| is huge */
                    185:        if(iy>0x41e00000) { /* if |y| > 2**31 */
                    186:            if(iy>0x43f00000){  /* if |y| > 2**64, must o/uflow */
                    187:                if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
                    188:                if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
                    189:            }
                    190:        /* over/underflow if x is not close to one */
1.13      drochner  191:            if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny;
                    192:            if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny;
1.11      simonb    193:        /* now |1-x| is tiny <= 2**-20, suffice to compute
1.1       jtc       194:           log(x) by x-x^2/2+x^3/3-x^4/4 */
1.13      drochner  195:            t = ax-one;         /* t has 20 trailing zeros */
1.1       jtc       196:            w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
                    197:            u = ivln2_h*t;      /* ivln2_h has 21 sig. bits */
                    198:            v = t*ivln2_l-w*ivln2;
                    199:            t1 = u+v;
1.4       jtc       200:            SET_LOW_WORD(t1,0);
1.1       jtc       201:            t2 = v-(t1-u);
                    202:        } else {
1.13      drochner  203:            double ss,s2,s_h,s_l,t_h,t_l;
1.1       jtc       204:            n = 0;
                    205:        /* take care subnormal number */
                    206:            if(ix<0x00100000)
1.4       jtc       207:                {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
1.1       jtc       208:            n  += ((ix)>>20)-0x3ff;
                    209:            j  = ix&0x000fffff;
                    210:        /* determine interval */
                    211:            ix = j|0x3ff00000;          /* normalize ix */
                    212:            if(j<=0x3988E) k=0;         /* |x|<sqrt(3/2) */
                    213:            else if(j<0xBB67A) k=1;     /* |x|<sqrt(3)   */
                    214:            else {k=0;n+=1;ix -= 0x00100000;}
1.4       jtc       215:            SET_HIGH_WORD(ax,ix);
1.1       jtc       216:
1.13      drochner  217:        /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
1.1       jtc       218:            u = ax-bp[k];               /* bp[0]=1.0, bp[1]=1.5 */
                    219:            v = one/(ax+bp[k]);
1.13      drochner  220:            ss = u*v;
                    221:            s_h = ss;
1.4       jtc       222:            SET_LOW_WORD(s_h,0);
1.1       jtc       223:        /* t_h=ax+bp[k] High */
                    224:            t_h = zero;
1.4       jtc       225:            SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
1.1       jtc       226:            t_l = ax - (t_h-bp[k]);
                    227:            s_l = v*((u-s_h*t_h)-s_h*t_l);
                    228:        /* compute log(ax) */
1.13      drochner  229:            s2 = ss*ss;
1.1       jtc       230:            r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
1.13      drochner  231:            r += s_l*(s_h+ss);
1.1       jtc       232:            s2  = s_h*s_h;
                    233:            t_h = 3.0+s2+r;
1.4       jtc       234:            SET_LOW_WORD(t_h,0);
1.1       jtc       235:            t_l = r-((t_h-3.0)-s2);
1.13      drochner  236:        /* u+v = ss*(1+...) */
1.1       jtc       237:            u = s_h*t_h;
1.13      drochner  238:            v = s_l*t_h+t_l*ss;
                    239:        /* 2/(3log2)*(ss+...) */
1.1       jtc       240:            p_h = u+v;
1.4       jtc       241:            SET_LOW_WORD(p_h,0);
1.1       jtc       242:            p_l = v-(p_h-u);
                    243:            z_h = cp_h*p_h;             /* cp_h+cp_l = 2/(3*log2) */
                    244:            z_l = cp_l*p_h+p_l*cp+dp_l[k];
1.13      drochner  245:        /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
1.1       jtc       246:            t = (double)n;
                    247:            t1 = (((z_h+z_l)+dp_h[k])+t);
1.4       jtc       248:            SET_LOW_WORD(t1,0);
1.1       jtc       249:            t2 = z_l-(((t1-t)-dp_h[k])-z_h);
                    250:        }
                    251:
1.14      christos  252:     /* split up y into yy1+y2 and compute (yy1+y2)*(t1+t2) */
                    253:        yy1  = y;
                    254:        SET_LOW_WORD(yy1,0);
                    255:        p_l = (y-yy1)*t1+y*t2;
                    256:        p_h = yy1*t1;
1.1       jtc       257:        z = p_l+p_h;
1.4       jtc       258:        EXTRACT_WORDS(j,i,z);
1.1       jtc       259:        if (j>=0x40900000) {                            /* z >= 1024 */
                    260:            if(((j-0x40900000)|i)!=0)                   /* if z > 1024 */
                    261:                return s*huge*huge;                     /* overflow */
                    262:            else {
                    263:                if(p_l+ovt>z-p_h) return s*huge*huge;   /* overflow */
                    264:            }
                    265:        } else if((j&0x7fffffff)>=0x4090cc00 ) {        /* z <= -1075 */
                    266:            if(((j-0xc090cc00)|i)!=0)           /* z < -1075 */
                    267:                return s*tiny*tiny;             /* underflow */
                    268:            else {
                    269:                if(p_l<=z-p_h) return s*tiny*tiny;      /* underflow */
                    270:            }
                    271:        }
                    272:     /*
                    273:      * compute 2**(p_h+p_l)
                    274:      */
                    275:        i = j&0x7fffffff;
                    276:        k = (i>>20)-0x3ff;
                    277:        n = 0;
                    278:        if(i>0x3fe00000) {              /* if |z| > 0.5, set n = [z+0.5] */
                    279:            n = j+(0x00100000>>(k+1));
                    280:            k = ((n&0x7fffffff)>>20)-0x3ff;     /* new k for n */
                    281:            t = zero;
1.4       jtc       282:            SET_HIGH_WORD(t,n&~(0x000fffff>>k));
1.1       jtc       283:            n = ((n&0x000fffff)|0x00100000)>>(20-k);
                    284:            if(j<0) n = -n;
                    285:            p_h -= t;
1.11      simonb    286:        }
1.1       jtc       287:        t = p_l+p_h;
1.4       jtc       288:        SET_LOW_WORD(t,0);
1.1       jtc       289:        u = t*lg2_h;
                    290:        v = (p_l-(t-p_h))*lg2+t*lg2_l;
                    291:        z = u+v;
                    292:        w = v-(z-u);
                    293:        t  = z*z;
                    294:        t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
                    295:        r  = (z*t1)/(t1-two)-(w+z*w);
                    296:        z  = one-(r-z);
1.4       jtc       297:        GET_HIGH_WORD(j,z);
1.1       jtc       298:        j += (n<<20);
                    299:        if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
1.4       jtc       300:        else SET_HIGH_WORD(z,j);
1.1       jtc       301:        return s*z;
                    302: }

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